Question: $f(x) = \dfrac{ 5 }{ \sqrt{ 7 - \lvert x \rvert } }$ What is the domain of the real-valued function $f(x)$ ?
Answer: First, we need to consider that $f(x)$ is undefined anywhere where the radicand (the expression under the radical) is less than zero. So we know that $7 - \lvert x \rvert \geq 0$ This means $\lvert x \rvert \leq 7$ , which means $-7 \leq x \leq 7$ Next, we need to consider that $f(x)$ is also undefined anywhere where the denominator is zero. So we know that $\sqrt{ 7 - \lvert x \rvert } \neq 0$ , so $\lvert x \rvert \neq 7$ This means that $x \neq 7$ and $x \neq -7$ So we have four restrictions: $x \geq -7$ $x \leq 7$ $x \neq -7$ , and $x \neq 7$ Combining these four, we know that $x > -7$ and $x < 7$ ; alternatively, that $-7 < x < 7$ Expressing this mathematically, the domain is $\{ \, x \in \RR \mid -7< x <7\, \}$.